Download Morita equivalence for regular algebras

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
F. G RANDJEAN
E. M. V ITALE
Morita equivalence for regular algebras
Cahiers de topologie et géométrie différentielle catégoriques, tome
39, no 2 (1998), p. 137-153
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C4HIERS DE TOPOLOlilE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
MORITA
I Volume XXXLX-2
(1998)
EQUIVALENCE FOR REGULAR ALGEBRAS
by F. GRANDJEAN and E.M.
VITALE
Resume: Nous 6tudions les categories des modules r6guliers
sur les algebres r6guli6res, afin de g6n6raliser certains r6sultats classiques de la th6orie de Morita, faits dans le cas des
algebres unitaires, au cas des algebres régulières.
Introduction: Classical Morita theory studies equivalences between
categories of unital modules over unital R-algebras, for R a commutative unital ring. The key result is the Eilenberg-Watts theorem, which
states that colimit preserving R-functors between module categories correspond to bimodules. Morita theory is also the base to define the
Brauer group of the ring R, which is the group of Morita equivalence
classes of Azumaya R-algebras.
The aim of this note is to study categories of regular modules over
not necessarily unital R-algebras, where a module M over an R-algebra
A is regular if the canonical morphism
induced
The
by the action of A over M, is an isomorphism (cf. [16]).
first, simple but crucial fact is that, if A itself is regular as
A-module, the category of regular A-modules is a colocalization of the
category of all A-modules. This allows us to prove the analogous of
the Eilenberg-Watts theorem and then to have a satisfactory Morita
theory for regular R-algebras. Our second result is that the (classifying
category of the bi-)category of regular R-algebras and regular bimodules
is compact closed. This gives us a quick construction of a group in which
the Brauer group of R embeds.
137
The
1
Eilenberg-Watts
theorem
We fix once for all a commutative unital ring R. Everything should
be intended as enriched over the category of unital R-modules. In particular, the unlabelled tensor product 0 is the tensor product over R,
and algebra means R-algebra. Modules and algebras are always associative, but not necessarily unital.
Definition 1.1 Let A be an algebra and M
that M is regular if the arrow
induced
by the
action A 0
M-&#x3E;M,
is
an
a
left A-module. We say
isomorphism.
We write A-mod for the category of left .4-modules and A-modreg
for its full subcategory of regular modules. In the same way one defines
the category modre9-A of regular right A-modules and the category Amod"9-B of bimodules which are regular both as left A-modules and
An algebra A is regular if it is regular as left
as right B-modules.
(equivalently, right) A-module.
Examples:
1) Clearly, if A is unital, then A is regular (a one-side unit is enough);
moreover, in this case a module is regular iff it is unital (cf. proposition 3.2 in
2)
-
-
-
Other
[17]).
examples of regular algebras
are:
rings with local units (cf. [1], [2], [3]) and, between them, rings of
infinite matrices with a finite number of non-zero entries;
left
or
right splitting algebras (cf. [17]);
separable algebras and,
unit
in
particular, Azumaya algebra without
(cf. [9], [10], [11], [14], [16]).
138
3)
and [10], regular bimodules over regular algebras are used
to define (strict) Morita contexts and then to give an algebraic
description of the second 6tale cohomology group of R.
4)
A general argument on coequalizers shows that the dual of a regular algebra is regular; the fact that the tensor product of two
regular algebras is regular will be proved in section 2.
In
[16]
Recall that each A-B-bimodule M induces
a
pair of adjoint functors
with M 0B - left adjoint to LinA(M, -). If M is regular as A-module,
then the functor M 0B - factors through A-mod"9 and, since A-modreg
is full in A-mod, we obtain an adjunction
In
particular,
Proposition
is
if A is
a
regular algebra,
1.2 Let A be
right adjoint
a
we
have
an
adjunction
regular algebra; the functor
to the full inclusion
Proof. let X be in A-modreg and Y in A-mod. Given an arrow
X -&#x3E;AXAY, we obtain an arrow X -&#x3E;AXAY -&#x3E;Y, where the second
component is the arrow induced by the action of A on Y. Conversely,
given an arrow g: X -&#x3E; Y, we obtain an arrow
where the first component
is the inverse of the arrow induced by the
A
on
action of
X. Precomposing with the isomorphism A 0A X-&#x3E;X,
one checks that these constructions are a bijection of hom-sets. The
naturality
is
obvious.
139
From the previous proposition, it follows that A-modreg is a colocalization of A-mod (that is, the full inclusion has a right exact right
adjoint). Since A-mod is a complete and cocomplete abelian category,
standard arguments on localizations (cf. [7] vol.1, ch.3; vol.2, ch.1) give
us the following:
Corollary 1.3 Let A be
complete, cocomplete and
a
regular algebra;
the category A-modreg is
abelian.
corollary allows us to have the "free " presentation of a regular
module, which is used to prove the Eilenberg-Watts theorem.
Lemma 1.4 Let A be a regular algebra and let X be in A-modre9; then
X is the coequalizer in A-modreg of a pair of arrows between copowers
This
of A.
Proof. consider the copower of A indexed
the canonical A-linear arrow
by the elements
of X and
Since X is
regular, the action A X X -&#x3E; X is surjective, and then also w
surjective; since A-modreg is abelian, p is the coequalizer of its kernel
pair
is
Now, repeat the argument starting from
We obtain the
following diagram,
N
which is
140
a
coequalizer
in A-modreg
functor F: B-mod-A-mod, with A and B two
arbitrary algebras. The A-module FB can be provided with a structure of right B-module (compatible with its structure of left A-module)
Consider
taking
the
as
arrow
now a
action
corresponding, by adjunction,
to the
composite
where the first component is induced by the multiplication of
second is the action of F and the third is the inclusion.
Definition 1.5 A functor F: B-mod-&#x3E;A-mod is
B-module (FB, MF) is regular.
Now
we are
ready
to state the
Eilenberg-Watts
B, the
regular if the right
theorem for
regular .
algebras.
Proposition
a
1.6 Let A and B be two
regular algebras
and
functor. The assignments
and
give rise to an adjoint equivalence between the category of regular and
colimit preserving functors
and the category of regular bimodules
Proof: given
M in
A-modreg-B,
the functor
141
preserves colimits because it factors
and then it is left
adjoint
to the functor
Via lemma 1.4, the rest of the
algebras (cf. [5]
or
as
proof runs as in the classical case of unital
[15]).
m
The previous proposition is a proper generalization of the EilenbergWatts theorem for unital algebras. In fact, the condition of regularity
on F always holds if B is unital. (More in general, it holds if F preserves
coproducts and if there exists a left B-linear arrow p: B---. llB B such
that
is a section for the multiplication, where 0 is the left B-linear
induced by tensoring with a fixed element of B.)
2
The
arrow
bicategory of regular algebras
In this section we generalize to regular algebras some classical consequences of the Eilenberg-Watts theorem. First of all, let us restate it in
the more meaningful language of bicategories (cf. [6]). We write Algreg
for the bicategory whose objects are regular algebras, whose 1-arrows
are regular bimodules and whose 2-arrows are morphisms of bimodules.
The composition of two bimodules M: A l-&#x3E; B and N: B l-&#x3E; C is their
tensor product M 0B N: A l-&#x3E; C. The identity on an object A is A itself
seen as A-A-bimodule. Proposition 1.6 can be now expressed in the
following way:
Proposition 2.1 The assignments of proposition 1.6 give rise to a
biequivalence between the 2-category of regular algebras and regular and
colimit-preserving functors, and the bicategory Algreg. This biequivalence is the identity on the objects.
Since any biequivalence preserves and reflects invertible I-arrows, we
have the following:
142
Corollary 2.2 The biequivalence of proposition 2.1 restricts to a biequivalence between regular equivalences and invertible regular bimodules.
In other words, two regular algebras are Morita equivalent (= there
exists a regular equivalence B-modreg -&#x3E; A-modreg) iff they are equivalent in the bicategory Algreg (= there exist two regular bimodules
M: A H B , N: B H A and two isomorphisms of bimodules M XB N =
A, NXAM = B).
As in the classical case, several Morita invariants
deduced (cf. [5] and [15]).
can
be
now
easily
2.3 Consider two Morita equivalent regular algebras A
and B. Let F: B-modre9 -&#x3E; A-modreg be the given regular equivalence,
M
FB the corresponding invertible regular A-B-bimodule, and N the
inverse of M.
Proposition
=
1)
there is
a
regular equivalence
2)
there is
a
strict monoidal
3)
the
4)
the centers of A and B
5)
the lattice of left regular ideals of B is
of regular A-submodules of M;
6)
the lattices of two-sided
regular Picard groups of A
Proof: 1)
induce
equivalence
and
are
and B
are
isomorphic;
isomorphic;
isomorphic
regular ideals of A
2):
and B
to the lattice
are
isomorphic.
the invertible 1-arrows M: A - B and N: B l-&#x3E; A
equivalences between hom-categories:
143
3): the regular Picard group Picreg (A) of A is the group of isomorphism
classes of regular autoequivalences of A-modreg. The equivalence
restricts to an equivalence between invertible regular B-B-bimodules
and invertible regular A-A-bimodules. We obtain then the isomorphism
Picreg (B) = Pic reg(A) localizing to B and A the biequivalence of Corollary 2.2.
4): consider once again the strict monoidal equivalence
in
particular,
that is
we
have
an
isomorphism
isomorphism between the centers of B and A, because TB is
isomorphic to A.
5) and 6): any equivalence induces an isomorphism between the lattice
of subobjects of an object and of its image. Considering the regular
equivalence
an
and the
sidering
B in B-mod"9, we have
the monoidal equivalence
and the
ob ject B
object
in
B-modre9-B,
we
point 5) of the
have
statement. Con-
point 6) of the
statement. m
The next two propositions, proved in [11], contain the main facts to
endow Algreg of a compact closed structure. They are quite obvious if
the algebras have units, but they need some more attention for regular
algebras.
144
Proposition 2.4 Let A and B be two algebras and consider M in
modre9-A, M’ in A-modre9, lV in modreg-B and N’ in B-modreg. The
obvious isomorphism
induces
an
Proof:
where
isomorphism
recall that M 0A M’ is
kA (m X a X m’) = nia 0 m’
given by the following quotient
- m X am’; analogously,
We need a pair of arrows a,B making commutative the
gram, where T: M’ 0 N- N 0 M’ is the twist
For
this, consider the inclusions
145-
we
have
following
dia-
and take the
images AA and AB of
Recall that
equivalence classes) is an isomorphism (cf. [8],
chap.2, §3, n.6). Now, instead of a and B, we can look for two arrows
a’, (3’ making commutative the following diagram
(square
brackets
are
To define a’, we have to show that, for each element p in AA + AB,
the element (1 (D T 0 1 ) (p) is in ImkAXB. Suppose first p of the form
p
(ma X m’ - m (9 am’ ) X n 0 n’ (that is, p is in AA). Since M and
N are regular, m and n can be written as
=
so
that
146
Finally, one checks
stracting the term
If p is in AB,
one
that this is
works in the
an
element of
same
way
ImÀAQ9B adding
and sub-
using the regularity of M’ and
N’.
Conversely, given an element y
in ImkAXB, we have
=
ma
0 nb X
m’ X n’ - m X n 0 am’ 0 bn’
which is in AA+XB. This allows us to define B’ and the
proof is complete.
D
2.5 Let A and B be two regular algebras and consider M
in modre9-A and N in modre9-B. Then A 0 B is a regular algebra and
M 0 N is in modre9-A (9 B.
Corollary
2.6 Let A and B be two regular algebras. The
A-modre9-B and A 0 Bop-modreg are isomorphic.
Proposition
Proof: we sketch the proof, more details
A-mod"9-B; we define an action
can
be found in
categories
[11].
Let
M be in
and
have to show that M is regular with respect to this action. This
essentially amount to show that
is
we
an
isomorphism. The crucial point
is to prove that its inverse
147
is well defined. This
means
B(a X a’m X b) = B(aa’ X m X b) , 0(a 0 mb’ 0 b) = B(a 0 m (g) b’b) .
We check the first condition, the second is similar: since A and B are
regular, we can write
so
that
Clearly, a morphism f : M-&#x3E; M’ in A-modre9-B is also A 0 BOP-linear
with respect to the action of A Q9 BOP on M and M’ just defined.
Conversely, consider N in A 0 Bop-modreg and let n be in N. Since
N is regular, we can write
and
we
define
148
In other
position,
action A X N-&#x3E; N is given
where mA is the multiplication of A :
words, the
by
the
following
com-
Tensorizing over A and using that A is regular, one has that N is regular
as left A-module. The argument for the action N 0 B - N is similar.
As far as the compatibility between the two actions is concerned, we
have
Now consider
regular
so
as
morphism f: N-N’
A-B-bimodule, we can write
a
in A X B’P-mod.
that
In the
same
way
one
proves that
f (nb)
149-
=
f (n)b.
Since N is
simple to verify that the two constructions just described are
mutually inverse, using once again that, if M is in A-modreg-B and N
is in A 0 Bop-modreg, we can write
It is
classifying category cl(B) of a bicategory B has been introduced
[6]: it has the same objects as B and, as arrows, 2-isomorphism classes
The
in
of 1-arrows of B. For an introduction to compact closed categories, the
reader can see [12]: they are symmetric monoidal categories in which
each object has a left adjoint.
Corollary
2.7 The category
cl(AlgTe9)
is compact closed.
Proof: by proposition 2.4, the tensor product of R-modules induces
product in cl(Algreg). Moreover, given three regular algebras
A, B and C, by proposition 2.6 we have a bijection
a
tensor
which in fact is natural in A and C. This
implies
that B°P is left
adjoint
to B (cf. [12]).
Recall that the Brauer group B(R) of the unital commutative ring R
is the group of Morita equivalence classes of unital Azumaya algebras.
Moreover, it is a known fact that a unital algebra is Azumaya if and
only if A X A°P is Morita equivalent to R (cf. [13], [20]). The previous
corollary allows us to embed the Brauer group into a bigger group built
up using regular algebras.
Proposition
2.8
1)
Morita equivalence classes of invertible regular algebras constitute
an abelian group in which B(R) embeds.
2)
A regular
to R.
algebra A
is invertible iff A 0 AOP is Morita
150
equivalent
Proof.- 1): by corollary 2.2, Morita equivalence classes of invertible
regular algebras are exactly the invertible elements of the commutative monoid of isomorphism classes of objects of cl(Algreg) (that is, the
monoid cl.(cl(Algreg))). The fact that B(R) embeds in this group follows
from the fact, already quoted, that a module on a unital algebra is regular iff it is unital.
2): if there exists a regular algebra B such that A X B is isomorphic, in
cl(Algre9), to R, -then B is left adjoint to A. But AOP is left adjoint to
A, so that B is isomorphic, in cl(Algreg), to A°p.
Remarks:
1)
In classical Morita theory,
induces an equivalence
one can
prove that if
a
bimodule M
then M is a faithfully projective A-module.
Because of the lack of projectivity of a not necessarily unital algebra, what remains true in our more general context is that M
is a generator, in the sense that the evaluation
(everything is unital),
is
2)
surjective.
In the first section we have proved that, if A is a regular algebra,
then A-modreg is a colocalization of A-mod. This implies that
A-modreg is abelian and then exact (in the sense of Barr, cf. [4]).
Moreover, by lemma 1.4, A is a (regular) generator in A-modreg .
By proposition 2.1 in [18], we deduce that A-modreg is a localization of the category of algebras of the monad 1r induced by the
adjunction
(where,
for each set
S’, Us A is the S-indexed copower of A) .
infinitary algebraic theory T, corresponding to the monad T,
151
The
fails
annular theory only for cardinality reasons (A is not
abstractly finite in A-mod"9). In fact, the category of algebras
of 1r is the exact completion of the full subcategory of A-modreg
spanned by copowers of A, and then it is abelian. This implies
that T X Z= T, where 0 is here the tensor product of theories
and Z is the theory of abelian groups (cf. [19)).
to be
an
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Françoise Grandjean
D6partement de Math6matiques
Universite Catholique de Louvain
Chemin du Cyclotron 2
1348 Louvain-la-Neuve, Belgium
[email protected]
Math. 103
(1982),
categories, J. Pure Appl.
Appl.
Enrico M. Vitale
Laboratoire LANGAL
Universite du Littoral
Quai Freycinet 1, B.P. 5526
59379 Dunkerque, France
[email protected]
153